reserve A,O for non empty set,
  R for Order of A,
  Ol for Equivalence_Relation of O,
  f for Function of O,A*,
  g for Function of O,A;

theorem Th1:
  OverloadedRSSign(#A,R,O,Ol,f,g #) is non empty non void reflexive
  transitive antisymmetric
proof
  set RS0 = OverloadedRSSign(#A,R,O,Ol,f,g #);
A1: field the InternalRel of RS0 = the carrier of RS0 by ORDERS_1:12;
  then
A2: the InternalRel of RS0 is_antisymmetric_in the carrier of RS0 by
RELAT_2:def 12;
  the InternalRel of RS0 is_reflexive_in the carrier of RS0 & the
InternalRel of RS0 is_transitive_in the carrier of RS0 by A1,RELAT_2:def 9
,def 16;
  hence thesis by A2,ORDERS_2:def 2,def 3,def 4;
end;
